Finite Group Databasebeta

← #16952999

Subgroups — #16952999

Order 4352.

500 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14528772C211yesmaximal, characteristic
2C4×#10587505C211yesmaximal, characteristic
3#14526909C214yesmaximal
4C17×#2563C211yesmaximal, characteristic
5C2×#10587505C2212yes—
6C2×#10587505C2212yes—
7#11762669—21no—
8C4×C22⋊Dic17—21no—
9#11762576—21no—
10#11762576—21no—
11C2×C17×C4⋊Q8C411yescharacteristic, complemented
12C2×C4×D4×C17C2211yescharacteristic
13C17×#393—21no—
14C17×#393—21no—
15C17×#390C2212yes—
16C17×#382C412yescomplemented
17C17×#380—21no—
18C17×#380—21no—
19C4×C4⋊C4×C17C411yescharacteristic, complemented
20C22×Q8×C17—21no—
21C22×D4×C17C2311yescharacteristic
22C23×C4×C17D411yescharacteristic
23C23×C4×C17D411yescharacteristic
24C17×C4⋊Q8—41no—
25C17×C4⋊Q8—41no—
26C4×D4×C17—41no—
27C4×D4×C17—41no—
28C2×C4⋊C4×C17—21no—
29C2×C4⋊C4×C17—21no—
30C2×C4⋊C4×C17C2×C412yes—
31C2×C4⋊C4×C17C2×C411yescharacteristic
32C2×C22⋊C4×C17—22no—
33C2×C22⋊C4×C17—22no—
34C2×C22⋊C4×C17D411yescharacteristic
35C2×C22⋊C4×C17D411yescharacteristic
36C2×C42×C17C2×C411yescharacteristic
37C2×C42×C17C2×C412yes—
38C2×C22⋊Dic17—22no—
39C2×C22⋊Dic17—22no—
40C2×C4⋊Dic17—41no—
41C2×C4×Dic17—41no—
42C17×#95—21no—
43C17×#95—22no—
44C17×#95—22no—
45#10587505—22no—
46#10587505—24no—
47#10587505Q814yescomplemented
48#10587502—41no—
49#10587502—41no—
50C24×C17C2×D411yescharacteristic
51C24×C17C2×D411yescharacteristic
52C2×Q8×C17—42no—
53C2×Q8×C17—82no—
54C2×D4×C17—41no—
55C2×D4×C17—41no—
56C2×D4×C17C2×Q812yes—
57C2×D4×C17#4112yes—
58C22×C4×C17—21no—
59C22×C4×C17—21no—
60C22×C4×C17—21no—
61C22×C4×C17—21no—
62C22×C4×C17—22no—
63C22×C4×C17—22no—
64C22×C4×C17—42no—
65C22×C4×C17—42no—
66C22×C4×C17C22⋊C411yescharacteristic
67C22×C4×C17C22⋊C411yescharacteristic
68C22×C4×C17C22⋊C411yescharacteristic
69C22×C4×C17C22⋊C411yescharacteristic
70C22×C4×C17C22×C411yescharacteristic
71C22×Dic17—42no—
72C22×Dic17—42no—
73C4⋊C4×C17—41no—
74C4⋊C4×C17—42no—
75C4⋊C4×C17—42no—
76C4⋊C4×C17—41no—
77C4⋊C4×C17—41no—
78C22⋊C4×C17—41no—
79C22⋊C4×C17—41no—
80C22⋊C4×C17—82no—
81C22⋊C4×C17—82no—
82C42×C17—21no—
83C42×C17—21no—
84C42×C17—42no—
85C22⋊Dic17—24no—
86C22⋊Dic17—42no—
87C22⋊Dic17—42no—
88C22⋊Dic17—42no—
89C4⋊Dic17—82no—
90C4×Dic17—82no—
91#13299—171nomaximal, Sylow
92C23×C17—21no—
93C23×C17—21no—
94C23×C17—21no—
95C23×C17—21no—
96C23×C17—41no—
97C23×C17—41no—
98C23×C17—41no—
99C23×C17—41no—
100C23×C17C2×C22⋊C411yescharacteristic
101C23×C17C22⋊Q812yes—
102C23×C17#12411yescharacteristic
103C23×C17C4⋊Q811yescharacteristic
104Q8×C17—84no—
105D4×C17—42no—
106D4×C17—42no—
107C2×C4×C17—21no—
108C2×C4×C17—21no—
109C2×C4×C17—22no—
110C2×C4×C17—21no—
111C2×C4×C17—21no—
112C2×C4×C17—21no—
113C2×C4×C17—42no—
114C2×C4×C17—42no—
115C2×C4×C17—41no—
116C2×C4×C17—41no—
117C2×C4×C17—41no—
118C2×C4×C17—42no—
119C2×C4×C17—42no—
120C2×C4×C17—42no—
121C2×C4×C17—42no—
122C2×C4×C17—42no—
123C2×C4×C17—42no—
124C2×C4×C17—81no—
125C2×C4×C17—81no—
126C2×C4×C17#9911yescharacteristic
127C2×C4×C17#9911yescharacteristic
128C2×C4×C17C4×Q812yes—
129C2×Dic17—44no—
130C2×Dic17—82no—
131C2×Dic17—82no—
132C2×Dic17—82no—
133C2×Dic17—82no—
134C2×Dic17—82no—
135#2563D1711yescharacteristic
136#1799—171no—
137C4×#99—171no—
138#1393—174no—
139C22×C17—21no—
140C22×C17—21no—
141C22×C17—22no—
142C22×C17—21no—
143C22×C17—21no—
144C22×C17—42no—
145C22×C17—41no—
146C22×C17—41no—
147C22×C17—41no—
148C22×C17—41no—
149C22×C17—41no—
150C22×C17—41no—
151C22×C17—81no—
152C22×C17—81no—
153C22×C17#39111yescharacteristic
154C22×C17C2×#9911yescharacteristic
155C22×C17#41011yescharacteristic
156C4×C17—21no—
157C4×C17—22no—
158C4×C17—41no—
159C4×C17—41no—
160C4×C17—41no—
161C4×C17—82no—
162C4×C17—82no—
163C4×C17#35412yes—
164Dic17—84no—
165Dic17—162no—
166C2×C4⋊Q8Dic1711yescharacteristic, complemented
167C2×C4×D4C2×D1711yescharacteristic
168C2×#99—172no—
169C2×#99—172no—
170#393—21no—
171#393—21no—
172#390C2×D1712yes—
173#382Dic1712yescomplemented
174#380—21no—
175#380—21no—
176#380—341no—
177C4×C4⋊C4Dic1711yescharacteristic, complemented
178C4×C22⋊C4—341no—
179#342—341no—
180#342—341no—
181C2×C17—21no—
182C2×C17—21no—
183C2×C17—42no—
184C2×C17—41no—
185C2×C17—41no—
186C2×C17#199611yescharacteristic
187C2×C17C2×#35411yescharacteristic
188C2×C17#212811yescharacteristic
189C22×Q8—21no—
190C22×D4C22×D1711yescharacteristic
191C23×C4C22⋊D1711yescharacteristic
192C23×C4C22⋊D1711yescharacteristic
193C4⋊Q8—41no—
194C4⋊Q8—41no—
195C4×D4—41no—
196C4×D4—41no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—681no—
200C2×C4⋊C4C2×Dic1712yes—
201C2×C4⋊C4C2×Dic1711yescharacteristic
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—342no—
205C2×C22⋊C4—342no—
206C2×C22⋊C4C22⋊D1711yescharacteristic
207C2×C22⋊C4C22⋊D1711yescharacteristic
208C2×C42—681no—
209C2×C42C2×Dic1711yescharacteristic
210C2×C42C2×Dic1712yes—
211#99—174no—
212#99—342no—
213#99—344no—
214#95—21no—
215#95—22no—
216#95—22no—
217#95—681no—
218#95—681no—
219C17#1329911yescharacteristic, complemented, Sylow
220C24C2×C22⋊D1711yescharacteristic
221C24C2×C22⋊D1711yescharacteristic
222C2×Q8—42no—
223C2×Q8—82no—
224C2×D4—41no—
225C2×D4—41no—
226C2×D4Q8×D1712yes—
227C2×D4#6325612yes—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—682no—
237C22×C4—682no—
238C22×C4C22⋊Dic1711yescharacteristic
239C22×C4C22⋊Dic1711yescharacteristic
240C22×C4C22⋊Dic1711yescharacteristic
241C22×C4C22⋊Dic1711yescharacteristic
242C22×C4C22×Dic1711yescharacteristic
243C4⋊C4—41no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—1362no—
249C22⋊C4—41no—
250C22⋊C4—41no—
251C22⋊C4—82no—
252C22⋊C4—82no—
253C22⋊C4—344no—
254C22⋊C4—682no—
255C22⋊C4—682no—
256C22⋊C4—682no—
257C42—21no—
258C42—21no—
259C42—42no—
260C42—1362no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23#1058763011yescharacteristic
270C23#1058763212yes—
271C23#1058763311yescharacteristic
272C23C2×C22⋊Dic1711yescharacteristic
273Q8—84no—
274D4—42no—
275D4—42no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—684no—
296C2×C4—1362no—
297C2×C4—1362no—
298C2×C4—1362no—
299C2×C4—1362no—
300C2×C4—1362no—
301C2×C4#1058750511yescharacteristic
302C2×C4#1058750511yescharacteristic
303C2×C4Q8×Dic1712yes—
304C22—21no—
305C22—21no—
306C22—22no—
307C22—21no—
308C22—21no—
309C22—41no—
310C22—42no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22—81no—
317C22—81no—
318C22#1176271611yescharacteristic
319C22C2×#1058750511yescharacteristic
320C22#1176278311yescharacteristic
321C4—21no—
322C4—22no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4—82no—
327C4—82no—
328C4—2722no—
329C4—1364no—
330C4#1176260412yes—
331C2—21no—
332C2—21no—
333C2—41no—
334C2—42no—
335C2—41no—
336C2#1452977011yescharacteristic
337C2C2×#1176260411yescharacteristic
338C2#1453045611yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC22×C4×C17C22×C4
FrattiniC22×D4C22×D17
FittingC17×#2563C2
socleC22×C17C2×#99
prime radical, cosocleC22×D4×C17C23
hypercenter#2563D17
hypocenterC17#13299
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2563

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
11. C2×C17×C4⋊Q8C4semidirect
16. C17×#382C4semidirect
19. C4×C4⋊C4×C17C4semidirect
47. #10587505Q8semidirect
166. C2×C4⋊Q8Dic17semidirect
173. #382Dic17semidirect
177. C4×C4⋊C4Dic17semidirect
219. C17#13299semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#145287722
C4×#105875052
#145269092
C17×#25632
#1329917